# The first and the last terms of an AP are 5 and 45 respectively. If the sum of all its terms is 400, Find its common difference.

**Given:** An A.P. , first term and last term 5 and 45 respectively. sum of all its terms$=400$.

**To do: **To find its common difference.

**Solution:**

Let $a$ be the first term and $d$ be the common difference.

Let us say the given A.P. has $n$ terms.

here as given,

first term $a=5$

last term $l=45$

sum of the A.P. $S_{n}=400$

$n=?$

As the given condition,

last term $l=a_{n}=a+(n-1)d$

$\Rightarrow 45=5+(n-1)d$

$\Rightarrow ( n-1) d=40\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .................( 1)$

And sum of all its terms $S_{n} =\frac{n}{2}[ 2a+( n-1) d]$

$\Rightarrow \frac{n}{2}[ 2\times 5+40] =400 \ \ \ \ \ ( \because ( n-1) d=40\ from\ ( 1)$

$\Rightarrow 50n=800$

$\Rightarrow n=\frac{800}{50} =16$

$n=16$, on subtituting the value of $n$ into $( 1)$ ,

$( 16-1) d=40$

$\Rightarrow 15d=40$

$\Rightarrow d=\frac{40}{15}$

$\Rightarrow d=\frac{8}{3}$

Therefore the common difference of the given A.P. is $\frac{8}{3} $.

- Related Articles
- The first and the last terms of an A.P. are 5 and 45 respectively. If the sum of all its terms is 400, find its common difference.
- The first and the last terms of an AP are 7 and 49 respectively. If sum of all its terms is 420, find its common difference.
- The first term of an A.P. is 5, the last term is 45 and the sum of all its terms is 400. Find the number of terms and the common difference of the A.P.
- The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
- The first term of an AP is \( -5 \) and the last term is 45 . If the sum of the terms of the AP is 120 , then find the number of terms and the common difference.
- The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
- If the sum of first p terms of an A.P., is $ap^{2} +bp$, find its common difference.
- The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?
- In an AP of 50 terms, the sum of first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the A.P.
- The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167. If the sum of the first ten terms of this AP is 235 , find the sum of its first twenty terms.
- The first and the last term of an A.P. are 8 and 350 respectively. If its common difference is 9, how many terms are there and what is their sum?
- The first and the last terms of an A.P. are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?
- The sum of first 6 terms of an \( \mathrm{AP} \) is 36 and the sum of its first 16 terms is 256. Find the sum of first 10 terms of this \( A P \).
- If the sum of first four terms of an A.P. is 40 and that of first 14 terms is 280. Find the sum of its first n terms.
- Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

##### Kickstart Your Career

Get certified by completing the course

Get Started